信号的时频分析-2

Time-dependent processing
1、Definition 2、Short-time energy 3、Short-time autocorrelation function
1、Definition A time-dependent measurement Tn on a signal x[n] is obtained by first applying a transformation T(.) (which is usually nonlinear) to the signal, and then lowpass filtering the transformed signal:
1、Definition 2、Time-dependent processing interpretation 3、Fourier transform interpretation
4、Filter-bank interpretation
5、Efficient implementation of the STFT using the FFT
aut signal xn[m]
Rn[0] = En if the same window w[n] is
used in both expressions.
The short-time autocorrelation function has local maxima when the lag k is a multiple of the period.
then making the change of variable l = m −n
3、Short-time autocorrelation function
A generalization of the short-time energy is the short-time autocorrelation function Rn[k], which is a function of two variables, time n and lag k.
The window w0[n] must be positive In the special case when w0[n] is a rectangular pulse of length 2N + 1 centered at the origin, this becomes :
signal energy over the interval [n−N, n+N].
The short-time autocorrelation function is useful for tracking the period of quasi-periodic signals (such as voiced speech) whose period slowly changes over time. Figures in next slide.
Short-time energy functions of a speech signal for Hamming windows of 5, 10, 20 and 40 ms.
（海明）
Long windows provide a smooth short-time energy, while short windows are best for resolving fine temporal variations in signal energy.
Because the autocorrelation function is a time average, the power spectrum represents the average frequency content of the signal over all times. These nonstationary signals are of indefinite duration, so that their Fourier transforms may not exist, and have statistical characteristics that change appreciably over time, so that it would not make sense to compute an average power spectrum for all times.
The second class of signals is that of stationary, random signals, which last indefinitely, and have stable statistical characteristics over long times.
C. Window implementation of the short-time energy.
D. Low-pass filter implementation of the short-time autocorrelation function.
E. Window implementation of the short-time autocorrelation function.
1、Definition
The short-time Fourier transform Xn(f) is function of two variables, time n and frequency f, which describes how the spectrum of restricted segments of a signal x[n] evolves with time. Formally, it is defined by:
While this definition may appear somewhat daunting, it can be interpreted from at least three different points of view.
A. Time-dependent processing interpretation of the STFT.
Short-time autocorrelation functions for (a) and (b) voiced speech, and (c) unvoiced speech using 40-ms Hamming windows.
The short-time Fourier transform
Ex: speech signals;EEG;EOG
we introduce techniques for processing a special class of nonstationary signals.
Introduction
The basic idea is to divide the signal into short time segments or “frames” over which the signal is approximately stationary, then make a set of measurements for each frame. Such time-dependent processing is applicable whenever the signal is quasi-stationary, i.e. when its statistical characteristics change slowly relative to the frame length. For example, speech can be considered to be quasistationary because the motions of the articulators are usually sufficiently slow that the spectral characteristics of speech change relatively little over intervals of 10 to 30 ms.
2、Short-time energy
A useful example of B is obtained when the transformation T(.) is a square function: The output signal is then the short-time energy En of the signal x[n]:
Class 1
Class 2
Ex: white noise
well-defined Fourier transforms power spectrum obtained by Fourier transforming the autocorrelation function
Introduction
Contents
Introduction
Time-dependent processing The short-time Fourier transform Applications of the short-time
Fourier transform
Introduction
Methods for two broad classes of signals: The first class is that of finite-energy signals.
Ex: Hamming windows
2πn w[n] 0.54 0.46 cos( ), 0 n N 1 N 1
Sometimes one is interested in the short-time
energy over a specific frequency band rather than that of the entire signal. In this case, the signal is first bandpass filtered, then the shorttime energy of the filter output is computed. This method is the best one when w0[n] is an IIR filter because the short-time energy can be evaluated recursively. On the other hand, if w0[n] is of finite duration, there exists an alternative method for computing the short-time energy.